Polytope of Type {4,12} with group SmallGroup(432,530)

Overview

Group: SmallGroup(432,530)
Rank: 3
Schläfli Type: {4,12}
Vertices, edges, …: 18, 108, 54
Order of s₀s₁s₂: 6
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

2-fold

{4,12}*216

3-fold

{4,4}*144

6-fold

{4,4}*72

54-fold

{2,2}*8

Covers minimal covers in bold

2-fold

{4,12}*864a

3-fold

4-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1*s2*s1)^2*s0*s1*s2> of order 2

27 facets

27 of {4}*8

9 vertex figures

9 of {12}*24

P/N, where N=<(s0*s1)^2> of order 2

30 facets

6 of {2}*4
24 of {4}*8

10 vertex figures

8 of {12}*24
2 of {6}*12

P/N, where N=<(s1*s0*s1*s2)^2> of order 3

18 facets

18 of {4}*8

6 vertex figures

6 of {12}*24

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^2*s0*s1*s2> of order 4

15 facets

3 of {2}*4
12 of {4}*8

5 vertex figures

4 of {12}*24
1 of {6}*12

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 6

12 facets

6 of {2}*4
6 of {4}*8

4 vertex figures

2 of {12}*24
2 of {6}*12

Representations

Permutation Representation (GAP)

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53);;
s1 := ( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)(17,23)(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)(43,51)(44,50)(45,49)(52,53);;
s2 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(54)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53);
s1 := Sym(54)!( 2, 3)( 4,10)( 5,12)( 6,11)( 7,19)( 8,21)( 9,20)(13,14)(16,24)(17,23)(18,22)(25,26)(29,30)(31,37)(32,39)(33,38)(34,46)(35,48)(36,47)(40,41)(43,51)(44,50)(45,49)(52,53);
s2 := Sym(54)!( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,34)( 8,36)( 9,35)(10,40)(11,42)(12,41)(13,37)(14,39)(15,38)(16,43)(17,45)(18,44)(19,49)(20,51)(21,50)(22,46)(23,48)(24,47)(25,52)(26,54)(27,53);
poly := sub<Sym(54)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1 >;

References

None.

to this polytope.